A394106 - OEIS

%I #5 Mar 10 2026 13:35:02

%S 5,4,2,6,5,9,2,8,1,4,2,7,2,2,9,0,7,4,5,0,1,1,1,1,8,7,2,5,8,1,7,7,2,6,

%T 7,1,6,5,7,1,6,7,3,2,6,0,2,4,9,5,4,3,5,1,1,2,0,9,0,7,3,7,9,8,9,4,2,2,

%U 6,4,4,1,0,5,0,1,7,4,1,0,6,2,3,4,4,2,7,5,1,2,3,2,8,1,8,3,6,1,6,1,0,2,9,9,4

%N Decimal expansion of the probability that three points uniformly and independently selected at random from the interior of a cube form the vertices of an obtuse triangle.

%H Dominik Beck, <a href="https://arxiv.org/abs/2501.11611">The Probability that a Random Triangle in a Cube is Obtuse</a>, arXiv:2501.11611 [math.MG], 2025.

%H Dominik Beck, <a href="https://www2.karlin.mff.cuni.cz/~beckd/lectures/DISSERTATION.pdf">Random polytopes</a>, doctoral thesis, Mathematical Institute of Charles University, Prague, 2025.

%F Equals 323338/385875 - 13*G/35 + 4859*Pi/62720 - 73*Pi/(1680*sqrt(2)) - Pi^2/105 + 3*Pi*log(2)/224 - 3*Pi*log(1 + sqrt(2))/224, where G is Catalan's constant (A006752).

%e 0.542659281427229074501111872581772671657167326024954...

%t RealDigits[323338/385875 - 13*Catalan/35 + 4859*Pi/62720 - 73*Pi/(1680*Sqrt[2]) - Pi^2/105 + 3*Pi*Log[2]/224 - 3*Pi*Log[1 + Sqrt[2]]/224, 10, 120][[1]]

%o (PARI) 323338/385875 - 13*Catalan/35 + 4859*Pi/62720 - 73*Pi/(1680*sqrt(2)) - Pi^2/105 + 3*Pi*log(2)/224 - 3*Pi*log(1 + sqrt(2))/224

%Y Cf. A006752.

%Y Related constants: A075549, A093072, A093588, A102519, A102520, A341942, A394107.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 10 2026